Factoring overview

Factoring is running multiplication backwards. To factor the integer 253, you consider how you could get it by multiplying smaller integers; the answer is that 253 = 11 × 23. Similarly, to factor the polynomial 2x² + 5x + 3, you consider how you could get it by multiplying simpler polynomials; the answer is that 2x² + 5x + 3 = (x + 1) ⋅ (2x + 3). (We'll discuss later how you could come up with that answer.) As the integers 11 and 23 are the factors of 253, so the polynomials x + 1 and 2x + 3 are the factors of 2x² + 5x + 3.

The simplest technique for factoring is when one of the factors is a constant. When you do this, here are some rules to follow, which apply whenever the original polynomial has only rational coefficients (otherwise it gets more complicated):

There can only be one constant factor, which comes first.
In each non-constant factor, the leading coefficient should be positive.
In each non-constant factor, all of the coefficients should be integers.
In each non-constant factor, the coefficients should not have any common integer factor greater than 1.

Here are some ways to factor 6x − 12 that violate these rules and so are wrong:

6x − 12 = 2(x − 2)3;
6x − 12 = −6(−x + 2);
6x − 12 = 12(½x − 1);
6x − 12 = 2(3x − 6).

All of these equations are identities (they're always true), but they are not the proper way to factor 6x − 12. The only proper way, following all of the rules above, is 6x − 12 = 6(x − 2).

Here are some more examples:

Look for a factor common to each term: 2y + 20 = 2 ⋅ y + 2 ⋅ 10 = 2(y + 10).
Sometimes it helps to factor the coefficients: 6t + 15 = 2 ⋅ 3 ⋅ t + 3 ⋅ 5 = 3(2t + 5).
You can also try to factor out a variable that appears in every term: 5x³ + 3x² = 5xxx + 3xx = x²(5x + 3).
A negative coefficient on the leading term means that a minus sign appears out front: 8 − 2x = −2x + 8 = −2x − −8 = −2(x − 4).
If the coefficients are fractional, then their common denominator appears in the denominator out front: ½x + ⅔ = 3/6 ⋅ x + 4/6 = ⅙(3x + 4).

Sometimes the textbook is a little sloppy about fractional and negative coefficients, and it's true that the rules are somewhat arbitrary; but you need to pick some rules and follow them consistently in order to guarantee that everything will work out in a complicated problem.

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This web page was written between 2012 and 2017 by Toby Bartels, last edited on 2017 August 24. Toby reserves no legal rights to it.

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